Existence And Multiplicity Of Solutions For Two Kinds Of Kirchhoff Type Equations

In this paper, we consider the Kirchhoff type problem where Ω2 (?) R3 is a bounded domain with regular boundary (?)Ω,1< q< 2, 4< p< 6, α,b, α1, b1 > 0, γ is a positive parameter, using the Mountain Pass Theorem and minimization arguments, we get that the problem above has at least three solutions: a nonnegative nontrivial solution and two nonpositive nontrivial solutions.Theorem 1 Suppose Ω (?) R3 is a bounded domain with regular boundary (?)Ω, and 1< q< 2,4< p< 6, α,b, α1 and b1> 0, α1> αγ1, γ1 is the positive principal eigenvalue of (-△, H01(Ω)), If γ is small enough, then problem (3) has at least three solutions: a nonnegative nontrivial solution and two nonpositive nontrivial solutions.Consider the following Kirchhoff type problem with critical exponent where Ω (?) R3 is a bounded domain with regular boundary (?)Ω,1< q< 2,4< p≤6, α, b, α1 and b1> 0, if γ is small enough,b1 is large enough, we get the existence of two nontrivial solutions for the problem above via the variational method.Theorem 2 Suppose Ω (?) R3 is a bounded domain with regular boundary (?)Ω and 1< q< 2,4< p≤ 6, α, b, α1 and b1> 0, α1 > αγ1, γ1 is the positive principal eigenvalue of (-△, H01(Ω)). If A is small enough, b1 is large enough, then problem (4) has at least three nontrivial solutions:a nonnegative nontrivial solution and two nonpositive nontrivial solutions.